Year of grant:
2004

Research Area:
Náttúra og -tilfeingi

Project type:
Ph.d.verkætlan

Project title:
Klassifikationsproblemer knyttet til symbolsk dynamik.

Grant number:
0401

Project manager:
Gunnar Restorff

Institution/company:
Institutt for Matematiske Fag, Københavns Universitet, Fróðskaparsetur Føroya og Kanada

Other participants:
Lektor Søren Eilers, Matematisk Afdeling, Institut for Matematiske Fag, Københavns Universitet.

Project period:
Original: 1.6.2005 - 9.8.2008
Final: 1.6.2005 - 9.8.2008

Grant from the FRC in DKK:
519.809 kr.

Project description:
Original description
Høvuðssendamálið við verkætlanini er, at fáa skil á tí generella klassifikatiónsprobleminum fyri reint óendaligar C*-algebrair – við útgangsstøði í høvuðursúrslitinum í mínari cand.scient.-serritgerð – og um høvi býðst, eisini at prógva eina meiri generellan klassifikatiónssetning fyri reint óendaligar C*-algebrair. Tað er okkara vón at kunna klassificera hesar C*-algebrair upp til stabila isomforfi og upp til eindarvarðveitandi isomorfi og harumframt eisini at klassificera automorfiirnar á hesum algebraum. Generelt finst fyri Cuntz-Krieger algebrair enn hvørki nøkur klassifkatión upp til eindarvarðveitandi isomorfi eiheldur nøkur klassifikatión av automorfiunum - klassifikatiónin upp til stabila isomorfi er lyst í framhaldi av mínari cand.scient.-serritgerð. Tað tykist sum ein gongd leið at nýta ein víðfevnandi isomorfisetning hjá Kirchberg – møguliga á sama hátt, sum Bonkat hevur gjørt í síni ritgerð.

Final report
The main theme of this thesis is classification of non-simple C*-algebras. It is based on the work obtained in the author’s Master’s thesis. The thesis is divided into two parts: a text-part and an appendices-part consisting of four papers.

The text-part contains two main results. The first one is a generalization of Bonkat’s Universal Coefficient Theorem (UCT) for Kirchberg’s ideal-related KK-theory with two specified ideals. Invoking results of Kirchberg, this gives classification of certain purely infinite C*-algebras with exactly two non-trivial ideals. The second result consists of a development of a notion of ideal-related K-theory with coefficients with one specified ideal, which has shown to be of relevance to classification of automorphisms of non-simple C*-algebras.

In the two first articles of the appendix we classify essential extensions of Kirchberg algebras and characterize the range of the invariants. In the third article C we classify certain extensions of algebras belonging to certain classes, where we can lift positive, invertible KK-elements to -isomorphisms. Also we have some applications of this to Matsumoto algebras and graph algebras. In the fourth article we give a series of examples showing that the naive guess of how to define ideal-related K-theory with coefficients does not work. Also we give an example showing that Bonkat’s UCT does not split, in general.

Project status:
Liðug

Project output:
Klassifikation af Cuntz-Krieger algebraer, speciale for cand.scient.-graden i matematik, Københavns Universitet, 2003. (udarbejdet før påbegyndelse af ph.d.-forløbet)

Classification of Cuntz-Krieger algebras up to stable isomorphism, Journal für die Reine und Angewandte Mathematik 298 (2006), pp. 185-210. (hovedsagligt udarbejdet før påbegyndelse af ph.d.-forløbet)

Classification of non-simple C*-algebras, Ph.D.-thesis in mathematics, University of Copenhagen, 2008.

[1] (with Søren Eilers) On Rørdam's classification of certain C*-algebras with one non-trivial ideal, in ”Procedings of the First Abel Symposium”, 2006.

[2] (with Efren Ruiz) On Rørdam's classification of certain C*-algebras with one non-trivial ideal, II, Mathematica Scandinavia 101(2) (2007), pp. 280-292.



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